Lecture 28: MON 23 MARLecture 28: MON 23 MARCh.30.10-11 Inductors & InductanceInductors & InductancePhysics 2102Jonathan DowlingNikolai TeslaEXAM 03: 6PM THU 02 APR 2009The exam will cover:Ch.28 (second half) throughCh.32.1-3 (displacement current,and Maxwell's equations).The exam will be based on:HW08 – HW110Consider a solenoid of length that has loops of area each, and windings per unit length. A current flows through the solenoid and generates a uniform magnetic field inNNA niB niµ==Inductance!!side the solenoid.The solenoid magnetic flux is .BNBA! =!B!( )20The total number of turns . The result we got for the special case of the solenoid is true for any inductor: . Here is a constant known as the of the solenoiBBN n n A iLi Lµ= ! " =" =! ! inductance2200d. The inductance depends on the geometry of the particular inductor. For the solenoid, .Bn AiL n Ai iµµ"= = =Inductance of the Solenoid!! L =µ0n2!A =µ0N / !( )2!A =µ0N2/ !ARL Circuit: Fluxing up an InductorRL Circuit: Fluxing up an Inductor• How does the currentin the circuit changewith time? !iR + E ! Ldidt= 0 i t( )=ER1 ! e! t /"( )Time constant of RL circuit: τ = L/RtE/Ri(t)Fast/Small τ Slow/Large τi(t)RL Circuit Fluxing UP i t( )=ER1 ! e! t /"( )i 0( )= 0 !i 0( )=EL VL0( )=E !i t( )=ELe"t /# VLt( )= L!i t( )= Ee"t /#!= L / R i !( )=ER!i "( )= 0VL!( )= 0Fluxing Down an InductorFluxing Down an Inductor0=+dtdiLiR i t( )=ERe! Rt / L=ERe! t /"tE/RExponential defluxingi(t)iThe switch is at a for a long time,until the inductor is charged. Then,the switch is closed to b.What is the current in the circuit?Loop rule around the new circuit:Inductors & EnergyInductors & Energy• Recall that capacitorsstore energy in an electricfield• Inductors store energy ina magnetic field. E = iR + Ldidt iE( )= i2R( )+ Lididt iE( )= i2R( )+ddtLi22!"#$%&Power delivered by battery= power dissipated by Ri+ (d/dt) energy stored in LP!=! iV!=!i2RInductors & EnergyInductors & EnergyP = LididtUB=Li22Magnetic Potential EnergyUB Stored in an Inductor.Magnetic Power Returnedfrom Fluxing DownInductor to Circuit or Putinto Fluxing Up Inductorfrom Circuit.ExampleExample• The switch has been inposition “a” for a long time.• It is now moved to position“b” without breaking thecircuit.• What is the total energydissipated by the resistoruntil the circuit reachesequilibrium?• When switch has been in position “a” for long time, currentthrough inductor = (9V)/(10Ω ) = 0.9A.• Energy stored in inductor = (0.5)(10H)(0.9A)2 = 4.05 J• When inductor “defluxes” through the resistor, all this storedenergy is dissipated as heat = 4.05 J.9 V10 Ω10 H!B!0Consider the solenoid of length and loop area that has windings per unit length. The solenoidcarries a current that generates a uniform magnetic field AniB niµ=Energy Density of a Magnetic Field! inside the solenoid. The magnetic fieldoutside the solenoid is approximately zero.2 2201The energy stored by the inductor is equal to .2 2This energy is stored in the empty space where the magnetic field is present.We define as energy density where is the volume BBBn A iU LiUu VVµ==!2 2 2 2 2 2 220 0 00 0inside the solenoid. The density .2 2 2 2This result, even though it was derived for the special case of a uniformmagnetic field, holds true in general.Bn A i n i n iBuAµ µ µµ µ= = = =!!20 2BBuµ=Units:!uB=[J/m3]Energy Density in E and B FieldsuE=!0E22uB=B22µ0The Energy Density of the Earth’s MagneticField Protects us from the Solar
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